Finite Gauge Theory on Fuzzy CP^2

TL;DR

This paper defines a non-perturbative, finite matrix model for U(n) gauge theory on fuzzy CP^2, capturing monopole and instanton solutions and connecting to classical Yang-Mills theory in the large N limit.

Contribution

It introduces a multi-matrix model for gauge theory on fuzzy CP^2, including explicit solutions and a finite, quantized path integral formulation.

Findings

Explicit monopole solutions on fuzzy CP^2

Finite N U(2) instanton solutions

Path integral quantization is finite

Abstract

We give a non-perturbative definition of U(n) gauge theory on fuzzy CP^2 as a multi-matrix model. The degrees of freedom are 8 hermitian matrices of finite size, 4 of which are tangential gauge fields and 4 are auxiliary variables. The model depends on a noncommutativity parameter 1/N, and reduces to the usual U(n) Yang-Mills action on the 4-dimensional classical CP^2 in the limit N -> \infty. We explicitly find the monopole solutions, and also certain U(2) instanton solutions for finite N. The quantization of the model is defined in terms of a path integral, which is manifestly finite. An alternative formulation with constraints is also given, and a scaling limit as R^4_\theta is discussed.